The viscous cahn-hilliard equation: morse decomposition and structure of the global attractor

Transactions of the American Mathematical Society

Volumn 351, Issue 6, 1999, Pages 2375-2406

  Grinfeld, M ^a  

^a UNIVERSITY OF STRATHCLYDE   (United Kingdom)

Author keywords

[No Author keywords available]

Indexed keywords

EID: 0001343091     PISSN: 00029947     EISSN: None     Source Type: Journal    
DOI: 10.1090/s0002-9947-99-02445-9     Document Type: Article

References (52)

1. N. Alikakos, P.W. Bates and G. Fusco, Slow motion for Cahn-Hilliard equation in one space dimension, J. Diff. Equations, 90 (1991), 81-134. MR 92a:35152
2. S. Allen and J.W. Cahn, A microscopic theory for antiphase boundary motion and its application to antiphase domain coarsening, Acta Metall. 27 (1979), 1084-1095.
3. N. Alikakos and G. Fusco, Equilibrium and dynamics of bubbles for the Cahn-Hilliard equation, Collection: Inter. Conf. on Differential Equations, vol 1,2 (Barcelona, 1991), 59-67,
4. World Sci. Publish., River Edge, NJ (1993). MR 95b:35111
5. N. Alikakos and G. Fusco, Slow dynamics for the Cahn-Hilliard equation in higher space dimensions. Part I: Spectral estimates, Comm. in PDEs 19 (1994), 1397-1447. MR 95j:35163
6. S. B. Angenent, The Morse-Smale property for a semilinear parabolic equation, JDE 62 (1986), 427-442. MR 87e:58115
7. F. Bai, C. M. Elliott, A. Gardiner, A. Spence and A. M. Stuart, The viscous Cahn-Hilliard equation. Part I: Computations, Nonlinearity 8 (1995), 131-160. MR 95m:35082
8. P. W. Bates and P. C. Fife, Spectral comparison principles for the Cahn-Hilliard and phase field equations, and time scales for coarsening, Physica 43D (1990), 335-348. MR 91h:80001
9. P. W. Bates and P. C. Fife, The dynamics of nucleation for the Cahn-Hilliard equation, SIAM J. Appl. Math, 53 (1993), 990-1008. MR 94g:82034
10. P. W. Bates and P. J. Xun, Metastable patterns for the Cahn-Hilliard equation, Part II, J. Diff. Equations, 117, (1995), 165-216. MR 95m:35188
11. D. Brochet, D. Hilhorst, and A. Novick-Cohen, Maximal attractor and inertial sets for a conserved phase field model, Advances in PDE 1(1996), 547-578. MR 97e:35189
12. L. Bronsard and D. Hilhorst, On the slow dynamics for the Cahn-Hilliard equation in one space dimension, Proc. Roy. Soc. London, Series A, 439 (1992), 669-682. MR 93k:35210
13. J.W. Cahn and J.E. Hilliard, Free energy of a nonuniform system. I. Interfacial free energy, J. Chem. Phys. 28 (1958), 258-267.
14. J. Carr, M. E. Gurtin and M. Slemrod, Structured phase transitions on a finite interval, Arch. Rat. Mech. Anal. 86 (1984), 317-354. MR 86i:80001
15. C. C. Conley, Isolated Invariant Sets and Morse Index, AMS Regional Conference Series in Mathematics 38 AMS, Providence, RI (1978). MR 80c:58009
16. A. Eden, C. Foias, B. Nicolaenko, and R. Temam, Inertial sets for dissipative evolution equations. Part I: Construction and application, IMA Preprint 812 (1991).
17. J. C. Eilbeck, J. E. Furter and M. Grinfeld, On a stationary state characterization of transition from spinodal decomposition to nucleation behavior in the Cahn-Hilliard model of phase separation, Phys. Lett. A 139 (1989), 42-46. MR 89k:80015
18. C. M. Elliott A. M. Stuart, The viscous Cahn-Hilliard equation. Part II: Analysis, J. Diff. Equations 128 (1996), 387-414. MR 97c:35080
19. C. M. Elliott and S. Zheng, On the Cahn-Hilliard equation, Arch. Rational Mech. Anal. 96 (1986), 339-357. MR 87k:80007
20. B. Fiedler, Global attractors of one-dimensional parabolic equations: 16 examples, Tatra Mount. Math. Publ. 4 (1994), 67-92. MR 95g:35089
21. B. Fiedler and C. Rocha, Heteroclinic orbits of semilinear parabolic equations, J. Diff. Equations 125, (1996) 239-281. MR 96k:58200
22. P.C. Fife, Models for phase separation and their mathematics, in Nonlinear Partial Differential Equations and Applications, M. Mimura and T. Nishida, eds., Kinokuniya Pubs., Tokyo, in press.
23. R. D. Franzosa, The continuation theory for Morse decompositions and connection matrices, Trans. AMS 310 (1988), 781-803. MR 90g:58111
24. R. D. Franzosa, The connection matrix theory for Morse decompositions, Trans. AMS 311 (1989), 561-592. MR 90a:58149
25. P. Freitas, Stability of stationary solutions for a nonlocal reaction-diffusion equation, Quart. J. Mech. Appl. Math., 48 (1995), 556-582. MR 97a:35111
26. G. Fusco and J.K. Hale, Slow motion manifolds, dormant instability and singular perturbations, Dynamics Differential Equations 1 (1989), 75-94. MR 90i:35131
27. M. Grinfeld, J. E. Furter and J. C. Eilbeck, A monotonicity theorem and its applications to stationary solutions of the phase field model, IMA J. Appl. Math. 49 (1992), 61-72; 50 (1993), 203-204. MR 93j:80003
28. M. Grinfeld and A. Novick-Cohen, Counting stationary solutions of the Cahn-Hilliard equation by transversality arguments, Proc. Royal Soc. Edin. A 125 (1995), 351-370. MR 96c:58157
29. M. E. Gurtin and H. Matano, On the structure of equilibrium phase transitions within the gradient theory of fluids, Quart. Appl. Math. 46 (1988), 301-317. MR 89j:49015
30. J. K. Hale, Asymptotic Behaviour of Dissipative Systems, Math. Surveys and Monographs 25 A.M.S. 1988. MR 89g:58059
31. H. Hattori and K. Mischaikow, A dynamical system approach to a phase transition problem, J. Diff. Equations 94 (1991), 340-378. MR 93m:58104
32. J. K. Hale and G. Raugel, Lower semicontinuity of attractors of gradient systems and applications, Ann. di Matem. Pura ed Applic. 154 (1989), 281-326. MR 91f:58087
33. D. Henry, Some infinite-dimensional Morse-Smale systems defined by parabolic differential equations, J. Diff. Equations 59 (1985), 165-205. MR 86m:58080
34. J. Mallet-Paret, Morse decompositions for delay-differential equations, J. Diff. Equations 72 (1988), 270-315. MR 80m:58112
35. H. Matano, Nonincrease of the lap number of a solution for a one-dimensional semilinear parabolic equation, J. Fac. Sci. Univ. Tokyo Sect. 1A Math. 29 (1982), 401-441. MR 84m:35060
36. K. Mischaikow, Global asymptotic dynamics of gradient-like bistable equations, SIAM J. Math. Analysis 26 (1995), 1199-1224. MR 96j:35025
37. B. Neithammer, Existence and uniqueness of radially symmetric stationary points within the gradient theory of phase transitions, Eur. J. Appl. Math. 6 (1995), 45-68.
38. B. Nicolaenko, B. Scheurer and R. Temam, Some global dynamical properties of a class of pattern formation equations, Comm. PDE 14 (1989), 245-297. MR 90e:35138
39. A. Novick-Cohen, The nonlinear Cahn-Hilliard equation: transition from spinodal decomposition to nucleation behavior, J. Stat. Phys. 38 (1985), 707-723.
40. A. Novick-Cohen, On the viscous Cahn-Hilliard equation, in: Material Instabilities in Continuum Mechanics and Related Mathematical Problems, J. M. Ball, ed., Oxford University Press, Oxford 1988. MR 89f:73005
41. A. Novick-Cohen, On Cahn-Hilliard type equations, Nonlinear Analysis TMA 15 (1990), 797-814. MR 92g:35105
42. A. Novick-Cohen, The Cahn-Hilliard equation: mathematical and modelling perspectives, Advances in Math. Sci. and Appl. 8 (1998), 965-985.
43. A. Novick-Cohen and L. A. Peletier, Steady states of the one-dimensional Cahn-Hilliard equation, Proc. Royal Soc. Edin. A 123 (1993), 1071-1098. MR 94m:35272
44. A. Novick-Cohen and L. A. Segel, Nonlinear aspects of the Cahn-Hilliard equation, Physica 10D (1984), 277-298. MR 85k:35120
45. I. Ohnishi and Y. Nishiura, Spectral comparison between the second and fourth order equations of conservative type with nonlocal terms, Japan J. Indust. Appl. Math. 15 (1998), 253-262.
46. J. Reineck, Connecting orbits in one-parameter families of flows, Erg. Theory Dynam. Sys. 8* (1988), 359-374. MR 89i:58128
47. J. Rubinstein and P. Sternberg, Non-local reaction-diffusion equations and nucleation, IMA J. Appl. Math. 48 (1992), 249-264. MR 93f:35116
48. R. Schaaf, Global Solution Branches of Two-Point Boundary Value Problems, Lecture Notes in Mathematics 1458, Springer-Verlag, Berlin 1990. MR 92a:34003
49. P. Sternberg and K. Zumbrun, Connectivity in phase boundaries in strictly convex domains, Arch. Rat. Mech. Anal. 141 (1998), 375-100.
50. R. Temam, Infinite-Dimensional Dynamical Systems in Mechanics and Physics, Applied Math. Sciences 68, Springer-Verlag, New York, 1988. MR 89m:35204
51. J. Wei and M. Winter, Stationary solutions for the Cahn-Hilliard equation, Ann. Inst. H. Poicare, Anal. Non Lin., 15 (1998), 459-192. CMP 98:15
52. S. Zheng, Asymptotic behaviour of solutions to the Cahn-Hilliard equation, Applic. Analysis 23 (1986), 165-184. MR 88b:35036